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This is a non-trivial question,
and the right way to do this is to go through the truth table,
which I've drawn over here.
There's 3 different things happening.
We've taken initial pick of the coin,
which can take coin 1 or coin 2 with equal probability,
and then you go flip it for the first time,
and there's heads or tails outcomes,
and we flip it for the second time with the second outcome.
So these different cases summarize my truth table.
I now need to observe just the cases where
head is followed by tail.
This one right here and over here.
Then we compute the probability for those 2 cases.
The probability of picking coin 1 is 0.5.
For the fair coin, we get 0.5 for heads,
followed by 0.5 for tails.
They're together is 0.125.
Let's do it with the second case.
There's a 0.5 chance of taking coin 2.
Now that one comes up with heads at 0.9.
It comes up with tails at 0.1.
So multiply these together, gives us 0.045,
a smaller number than up here.
Adding these 2 things together results in 0.17,
which is the right answer
to the question over here.
That was really non-trivial,
and I'd be amazed if you got this correct.